swarm ui: the login button's wording is a tested function

The label that switches between log in and log in again moves next to the
stored login's other wording, so it has a test.

frontend/packages/swarm-ui/src/agent/claude-login/StoredLogin.ts

@@ -24,6 +24,13 @@ export function storedAtMs(login: StoredLogin): number | null {
   return Number.isNaN(ms) ? null : ms;
 }
 
+// The settings row's login button: `log in` when the swarm holds no login for
+// the agent, `log in again` when it holds one, so a forced relogin reads as
+// what it is.
+export function loginButtonLabel(login: StoredLogin): string {
+  return login.stored_at === null ? "log in" : "log in again";
+}
+
 // What the delete's confirmation says: it removes the swarm's copy, and the
 // agent reads the store every 5 minutes and logs out when the copy is gone.
 export function deleteMessage(agent: string): string {

frontend/packages/swarm-ui/src/agent/detail/AgentClaudeLogin.tsx

@@ -17,6 +17,7 @@ import { showToast } from "../../ui/toast/toasts.js";
 import {
   deleteMessage,
   deleteStoredLoginPath,
+  loginButtonLabel,
   storedAtMs,
   storedLoginPath,
   type StoredLogin,
@@ -81,7 +82,7 @@ export function AgentClaudeLogin({ agent }: { agent: string }) {
   const loginBadge = (
     <Badge
       variant="quiet"
-      value={login.stored_at === null ? "log in" : "log in again"}
+      value={loginButtonLabel(login)}
       title={`log ${agent} in to claude from here`}
       onClick={() => setLoggingIn(true)}
     />

frontend/packages/swarm-ui/test/stored-login.test.mjs

@@ -6,6 +6,7 @@ import { test } from "node:test";
 import {
   deleteMessage,
   deleteStoredLoginPath,
+  loginButtonLabel,
   storedAtMs,
   storedLoginPath,
 } from "../src/agent/claude-login/StoredLogin.ts";
@@ -32,3 +33,11 @@ test("the delete's confirmation names the agent", () => {
   assert.match(deleteMessage("atlas"), /stored claude login for atlas/);
   assert.match(deleteMessage("atlas"), /logged out within 5 minutes/);
 });
+
+test("the login button says log in until a login is stored, then log in again", () => {
+  assert.equal(loginButtonLabel({ stored_at: null }), "log in");
+  assert.equal(
+    loginButtonLabel({ stored_at: "2026-11-10T12:00:00Z" }),
+    "log in again",
+  );
+});